Hence, for this particular scenario, the ratio of \(| \vec{E}^+(x) |\big |_{\rm min} /| \vec{E}^+(x) |\big |_{\rm max}\)  determining the ellipticity angle \(\chi \)  could be evaluated along the same lines as in the explicit calculation performed in Sec. REF  above.
This would result in a coordinate-dependent ellipticity angle [1]}. The ellipticity angle associated with the beam could, e.g., be defined in terms of the average of this ratio over the beam cross-section at a given longitudinal position in the far field.
